I'm looking for a iso function that maps reals to positive reals the inverse also needs to be $C^1$. But it need it needs to be efficient to be calculated on computers. Using maps like
$$ f(x)= \begin{cases} \dfrac{1}{x} - 2 & \text{if}\ 0 < x \leq \dfrac{1}{2}\\ \dfrac{1}{x-1} + 2 & \text{if} \ \dfrac{1}{2} < x < 1 \end{cases} $$
Do not work very well since when $x$ approaches the interval bounds $0, 1$, the function explodes and goes quickly to infinity, taking too much memory space for small increases.
Also I need it to be $C^1$ since I'm looking for their derivatives.